The Riemann invariant \(R_{-}\)  on the characteristic \(C_{-}\)  given by () of the non-dispersive equations is
conserved through the DSW [1]}, [2]}. As \(\rho = u_{+}^{2}\)  and \(v=0\)  at the leading edge of the DSW,
\(R_{-} = -2\sqrt{2}u_{+}/\sqrt{q} = v_{i} - 2\sqrt{2}u_{i}/\sqrt{q}\) , so that on using expression (REF )
from the expansion wave solution we have [3]}, [4]}.
\(u_{i} = \frac{1}{2}\left( u_{-} + u_{+} \right), \quad v_{i} = \frac{\sqrt{2}}{\sqrt{q}} \left( u_{-} - u_{+} \right).\) 
